Nittiya Pabhapote

Group Divisible Designs with Two Associate Classes and 2=1

The original classiffcation of PBIBDs defined group divisible designs GDD( 𝑣 = 𝑣 1 + 𝑣 2 + ⋯ + 𝑣 𝑔 , 𝑔 , 𝑘 , 𝜆 1 , 𝜆 2 ) with 𝜆 1 ≠ 0 . In this paper, we prove that the necessary conditions are suffcient for the existence of the group divisible designs with two groups of unequal sizes and block size three with 𝜆 2 = 1 .

Ramanujan sums via generalized Möbius functions and applications

A generalized Ramanujan sum (GRS) is defined by replacing the usual Mobius function in the classical Ramanujan sum with the Souriau-Hsu-Mobius function. After collecting basic properties of a GRS, mostly containing existing ones, seven aspects of a GRS are studied. The first shows that the unique representation of even functions with respect to GRSs is possible. The second is a derivation of the mean value of a GRS. The third establishes analogues of the remarkable Ramanujans formulae connecting divisor functions with Ramanujan sums. The fourth gives a formula for the inverse of a GRS. The fifth is an analysis showing when a reciprocity law exists. The sixth treats the problem of dependence. Finally, some characterizations of completely multiplicative function using GRSs are obtained and a connection of a GRS with the number of solutions of certain congruences is indicated.

Completely multiplicative functions arising from simple operations

Given two multiplicative arithmetic functions, various conditions for their convolution, powers, and logarithms to be completely multiplicative, based on values at the primes, are derived together with their applications.

GDDs with Two Associate Classes and with Three Groups of Sizes 1, n, n and λ1<λ2

A group divisible design GDDv=1+n+n,3, 3, λ 1, λ 2 is an ordered pair

Combinatorial Aspects of the Generalized Euler's Totient

V, \mathcal{B}

Logarithmic Operators and Characterizations of Completely Multiplicative Functions

where V is an 1+n+n-set of symbols and

Properties of rational arithmetic functions

\mathcal{B}

Group Divisible Designs with Two Associate Classes and λ2 1

is a collection of 3-subsets called blocks of V satisfying the following properties: the 1+n+n-set is divided into 3 groups of sizes 1, n and n; each pair of symbols from the same group occurs in exactly λ 1 blocks in

Distributive property of completely multiplicative functions

\mathcal{B}

GDDs with two associate classes and with three groups of sizes 1, n and n.

; and each pair of symbols from different groups occurs in exactly λ 2 blocks in